Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else - Hardcover

Ellenberg, Jordan

Hardcover

ISBN 10: 1984879057 ISBN 13: 9781984879059

Verlag: Penguin Press, 2021

Inhaltsangabe

An instant New York Times Bestseller!

“Unreasonably entertaining . . . reveals how geometric thinking can allow for everything from fairer American elections to better pandemic planning.” —The New York Times  

From the New York Times-bestselling author of How Not to Be Wrong—himself a world-class geometer—a far-ranging exploration of the power of geometry, which turns out to help us think better about practically everything.

How should a democracy choose its representatives? How can you stop a pandemic from sweeping the world? How do computers learn to play Go, and why is learning Go so much easier for them than learning to read a sentence? Can ancient Greek proportions predict the stock market? (Sorry, no.) What should your kids learn in school if they really want to learn to think? All these are questions about geometry. For real.

If you're like most people, geometry is a sterile and dimly remembered exercise you gladly left behind in the dust of ninth grade, along with your braces and active romantic interest in pop singers. If you recall any of it, it's plodding through a series of miniscule steps only to prove some fact about triangles that was obvious to you in the first place. That's not geometry. Okay, it is geometry, but only a tiny part, which has as much to do with geometry in all its flush modern richness as conjugating a verb has to do with a great novel.

Shape reveals the geometry underneath some of the most important scientific, political, and philosophical problems we face. Geometry asks: Where are things? Which things are near each other? How can you get from one thing to another thing? Those are important questions. The word "geometry"comes from the Greek for "measuring the world." If anything, that's an undersell. Geometry doesn't just measure the world—it explains it. Shape shows us how.

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Über die Autorin bzw. den Autor

Jordan Ellenberg is the John D. MacArthur Professor of Mathematics at the University of Wisconsin-Madison. His writing has appeared in Slate, The Wall Street Journal, The New York Times, The Washington Post, Wired, and the Believer.

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Chapter 1

 

"I Vote for Euclid"

 

In 1864, the Reverend J. P. Gulliver, of Norwich, Connecticut, recalled a conversation with Abraham Lincoln about how the president had acquired his famously persuasive rhetorical skill. The source, Lincoln said, was geometry.

 

In the course of my law-reading I constantly came upon the word demonstrate. I thought, at first, that I understood its meaning, but soon became satisfied that I did not. . . . I consulted Webster's Dictionary. That told of "certain proof," "proof beyond the possibility of doubt;" but I could form no idea what sort of proof that was. I thought a great many things were proved beyond a possibility of doubt, without recourse to any such extraordinary process of reasoning as I understood "demonstration" to be. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man. At last I said, "Lincoln, you can never make a lawyer if you do not understand what demonstrate means;" and I left my situation in Springfield, went home to my father's house, and staid there till I could give any propositions in the six books of Euclid at sight. I then found out what "demonstrate" means, and went back to my law studies.

 

Gulliver was fully on board, replying, "No man can talk well unless he is able first of all to define to himself what he is talking about. Euclid, well studied, would free the world of half its calamities, by banishing half the nonsense which now deludes and curses it. I have often thought that Euclid would be one of the best books to put on the catalogue of the Tract Society, if they could only get people to read it. It would be a means of grace." Lincoln, Gulliver tells us, laughed and agreed: "I vote for Euclid."

 

Lincoln, like the shipwrecked John Newton, had taken up Euclid as a source of solace at a rough time in his life; in the 1850s, after a single term in the House of Representatives, he seemed finished in politics and was trying to make a living as an ordinary traveling lawyer. He had learned the rudiments of geometry in his earlier job as a surveyor and now aimed to fill the gaps. His law partner William Herndon, who often had to share a bed with Lincoln at small country inns in their sojourns around the circuit, recalls Lincoln's method of study; Herndon would fall asleep, while Lincoln, his long legs hanging over the edge of the bed, would stay up late into the night with a candle lit, deep in Euclid.

 

One morning, Herndon came upon Lincoln in their offices in a state of mental disarray:

He was sitting at the table and spread out before him lay a quantity of blank paper, large heavy sheets, a compass, a rule, numerous pencils, several bottles of ink of various colors, and a profusion of stationery and writing appliances generally. He had evidently been struggling with a calculation of some magnitude, for scattered about were sheet after sheet of paper covered with an unusual array of figures. He was so deeply absorbed in study he scarcely looked up when I entered.

 

Only later in the day did Lincoln finally get up from his desk and tell Herndon that he had been trying to square the circle. That is, he was trying to construct a square with the same area as a given circle, where to "construct" something, in proper Euclidean style, is to draw it on the page using just two tools, a straightedge and a compass. He worked at the problem for two straight days, Herndon remembers, "almost to the point of exhaustion."

 

I have been told that the so-called squaring of the circle is a practical impossibility, but I was not aware of it then, and I doubt if Lincoln was. His attempt to establish the proposition having ended in failure, we, in the office, suspected that he was more or less sensitive about it and were therefore discreet enough to avoid referring to it.

 

Squaring the circle is a very old problem, whose fearsome reputation I suspect Lincoln might actually have known; "squaring the circle" has been a metaphor for a difficult or impossible task for a long time. Dante name-checks it in the Paradiso: "Like the geometer who gives his all trying to square the circle, and still can't find the idea he needs, that's how I was." In Greece, where it all started, a standard exasperated comment when someone is making a task harder than necessary is to say, "I wasn't asking you to square the circle!"

 

There is no reason one needs to square a circle-the problem's difficulty and fame is its own motivation. People with a conquering mentality tried to square circles from antiquity until 1882, when Ferdinand von Lindemann proved it couldn't be done (and even then a few die-hards persisted; okay, even now). The seventeenth-century political philosopher Thomas Hobbes, a man whose confidence in his own mental powers is not fully captured by the prefix "over," thought he'd cracked it. Per his biographer John Aubrey, Hobbes discovered geometry in middle age and quite by accident:

 

Being in a Gentleman's Library, Euclid's Elements lay open, and 'twas the 47 El. Libri 1. He read the Proposition. By G_, saydd he (he would now and then sweare an emphaticall Oath by way of emphasis) this is impossible! So he reads the Demonstration of it, which referred him back to such a Proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps that at last he was demonstratively convinced of that trueth. This made him in love with Geometry.

 

Hobbes was constantly publishing new attempts and getting in petty feuds with the major British mathematicians of the time. At one point, a correspondent pointed out that one of his constructions was not quite correct because two points P and Q he claimed to be equal were actually at very slightly different distances from a third point R; 41 and about 41.012 respectively. Hobbes retorted that his points were big enough in extent to cover such a minor difference. He went to his grave still telling people he'd squared the circle.

 

An anonymous commentator in 1833, reviewing a geometry textbook, described the typical circle-squarer in a way that quite precisely depicts both Hobbes, two centuries prior, and intellectual pathologies still hanging around here in the twenty-first:

 

[A]ll they know of geometry is, that there are in it some things which those who have studied it most have long confessed themselves unable to do. Hearing that the authority of knowledge bears too great a sway over the minds of men, they propose to counterbalance it by that of ignorance: and if it should chance that any person acquainted with the subject has better employment than hearing them unfold hidden truths, he is a bigot, a smotherer of the light of truth, and so forth.

 

In Lincoln, we find a more appealing character: enough ambition to try, enough humility to accept that he hadn't succeeded.

 

What Lincoln took from Euclid was the idea that, if you were careful, you could erect a tall, rock-solid building of belief and agreement by rigorous deductive steps, story by story, on a foundation of axioms no one could doubt: or, if you like, truths one holds to be self-evident. Whoever doesn't hold those truths to be self-evident is excluded from discussion. I hear the echoes of Euclid in Lincoln's most famous speech, the Gettysburg Address, where he characterizes the United States as "dedicated to the proposition that all men are created equal." A "proposition" is the term Euclid uses for a fact that follows logically from the self-evident...

Details zur Ausgabe

• Verlag: Penguin Press
• Erscheinungsdatum: 2021
• Sprache: Englisch
• ISBN 10: 1984879057
• ISBN 13: 9781984879059
• Einband: Gebundene Ausgabe
• Anzahl der Seiten: 480
• Kontakt zum Hersteller: Nicht verfügbar
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